# Full text of "Treatise On Analysis Vol-Ii"

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```92       XII    TOPOLOGY AND TOPOLOGICAL ALGEBRA

4.    Let E be a separable metrizable space. For each subset A of E, let D(A) denote the set
of points x e E such that, for each neighborhood V of x, the set V n A is not meager.
We have D(A) <= A. Show that D(A) = 0 if and only if A is meager. Show that
D(A) is closed and that A n (E — D(A)) is meager. Show that D(A) is the closure
of its interior. (If D'(A)  is  the closure of the  interior of D(A),  show  that
A n (E - D'(A)) is meager.)

5.    Let G be a separable metrizable group. Show that if H is a nonmeager subgroup of G,
then H is an open subgroup of G. (With the notation of Problem 4, show that the
interior of D(H) is not empty.)

6.    If a complete metrizable group is denumerable, then it is discrete.

7.    Let G be a separable, metrizable, locally compact group, and let H be a closed normal
subgroup of G. Show that the canonical bijection A/(A n H)-*AH/H is an iso-
morphism of topological groups.

8.    Let E, F, G be three metric spaces and d, d', d" the distances on E, F, G, respectively.
Let/be a mapping of E x F into G such that for each x0 e E the mapping .yi—»/(•*<>, y)
is continuous on F, and for each y0 e F the mapping x\-+f(x, y0) is continuous on E.

(a)    For each e > 0, each b e F and each x e E, let g(x; b, e) be the least upper bound
of the set of numbers a > 0 such that d'(b, y) g a implies d"(f(xt b), f(x, y)) <J e.
Show that the mapping x\-*g(x; b, e) is upper semicontinuous.

(b)    If E is complete, deduce from (a) that for each b e F there exists a meager set
Ma c E such that/is continuous at (a, b) for all a \$ M&.

If F also is complete, deduce that there exists a meager set N in E x F such that
the restriction of/to (E x F) — N is continuous. (For each A:, consider the set of points
at which the oscillation of/is < 1/&, and use Problem 4 of Section (12.7).

(c)    If E, F, G are Frechet spaces and/is a bilinear mapping of E x F into G, such
that the linear mappings f(x0t •) and /(•, y0) are continuous for all XQ € E and all
y0 e F, show that /is continuous on E x F.

9.    Let G be a group and d a distance on G defining a topology for which G is a complete,
separable, locally compact, metric space. Suppose that for each x0 e G the mappings
yt-+x0y and y\-^-yx0 are continuous on G. Show that the topology defined by d is
compatible with the group structure of G. (First prove that the mapping (x, y)t-+xy
is continuous, by using Problem 8. Then consider the set F in G x G consisting of all
(x3 x"1) with x e G. Show that F, with the law of composition (x, x~"l)(y, y~l) —
(xy, y"~1x~1) and the topology induced by the product topology on G x G, is a
topological group and is closed in G x G. Finally show that the restriction of pri
to F is bicontinuous, by using (12.16.12), and deduce that the mapping x\-^x"~1 is
continuous on G.)

10. Let T be a metric space, E a Frechet space, and M a set of mappings of E x T into a
metrizable topological vector space F, satisfying the following conditions: (1) for
each t0 e T, the set of mappings /(•, /o), where /e M, is an equicontinuous set of
linear mappings of E into F; (2) for each x0 e E, the set of mappings f(x0, •)> where
/e M, is equicontinuous. Show that M is equicontinuous. (Given t0 e T and a
balanced closed neighborhood V of 0 in F, for each x e E let dx be the least upper
bound of radii of open balls with centre (0 in T, such that for every point t of such a
ball we have f(x, f) —/(#, /0) e V for all /e M. Show that the mapping x\-^dx is
upper semicontinuous on E (argue by contradiction); then use (12.16.2).)on. The weak topology on
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